Tessellation technique in solving the two-point boundary value problem in multidimensional billiard spaces

Abstract

Two-point second order boundary value problem x¨(t)=f(t,x(t)), x(0)=A, x(T)=B in an n-dimensional polyhedron K with absolutely elastic impacts on the boundary of K is considered. The existence and multiplicity of solutions, especially having first impacts only at (n−1)-dimensional faces of K, is proved. The notion of an admissible billiard space is provided, and the unfolding technique is used for such spaces to reformulate the problem into a non-impulsive problem on Rn. As an illustration of the unfolding method and main results, the model of a system of two colliding balls on uneven ground is presented.